Step | Hyp | Ref
| Expression |
1 | | lmhmfgima.y |
. 2
β’ π = (π βΎs (πΉ β π΄)) |
2 | | lmhmfgima.xf |
. . . 4
β’ (π β π β LFinGen) |
3 | | lmhmfgima.f |
. . . . . 6
β’ (π β πΉ β (π LMHom π)) |
4 | | lmhmlmod1 20510 |
. . . . . 6
β’ (πΉ β (π LMHom π) β π β LMod) |
5 | 3, 4 | syl 17 |
. . . . 5
β’ (π β π β LMod) |
6 | | lmhmfgima.a |
. . . . 5
β’ (π β π΄ β π) |
7 | | lmhmfgima.x |
. . . . . 6
β’ π = (π βΎs π΄) |
8 | | lmhmfgima.u |
. . . . . 6
β’ π = (LSubSpβπ) |
9 | | eqid 2737 |
. . . . . 6
β’
(LSpanβπ) =
(LSpanβπ) |
10 | | eqid 2737 |
. . . . . 6
β’
(Baseβπ) =
(Baseβπ) |
11 | 7, 8, 9, 10 | islssfg2 41427 |
. . . . 5
β’ ((π β LMod β§ π΄ β π) β (π β LFinGen β βπ₯ β (π«
(Baseβπ) β©
Fin)((LSpanβπ)βπ₯) = π΄)) |
12 | 5, 6, 11 | syl2anc 585 |
. . . 4
β’ (π β (π β LFinGen β βπ₯ β (π«
(Baseβπ) β©
Fin)((LSpanβπ)βπ₯) = π΄)) |
13 | 2, 12 | mpbid 231 |
. . 3
β’ (π β βπ₯ β (π« (Baseβπ) β© Fin)((LSpanβπ)βπ₯) = π΄) |
14 | | inss1 4193 |
. . . . . . . . . 10
β’
(π« (Baseβπ) β© Fin) β π«
(Baseβπ) |
15 | 14 | sseli 3945 |
. . . . . . . . 9
β’ (π₯ β (π«
(Baseβπ) β© Fin)
β π₯ β π«
(Baseβπ)) |
16 | 15 | elpwid 4574 |
. . . . . . . 8
β’ (π₯ β (π«
(Baseβπ) β© Fin)
β π₯ β
(Baseβπ)) |
17 | | eqid 2737 |
. . . . . . . . 9
β’
(LSpanβπ) =
(LSpanβπ) |
18 | 10, 9, 17 | lmhmlsp 20526 |
. . . . . . . 8
β’ ((πΉ β (π LMHom π) β§ π₯ β (Baseβπ)) β (πΉ β ((LSpanβπ)βπ₯)) = ((LSpanβπ)β(πΉ β π₯))) |
19 | 3, 16, 18 | syl2an 597 |
. . . . . . 7
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (πΉ β ((LSpanβπ)βπ₯)) = ((LSpanβπ)β(πΉ β π₯))) |
20 | 19 | oveq2d 7378 |
. . . . . 6
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (π βΎs (πΉ β ((LSpanβπ)βπ₯))) = (π βΎs ((LSpanβπ)β(πΉ β π₯)))) |
21 | | lmhmlmod2 20509 |
. . . . . . . . 9
β’ (πΉ β (π LMHom π) β π β LMod) |
22 | 3, 21 | syl 17 |
. . . . . . . 8
β’ (π β π β LMod) |
23 | 22 | adantr 482 |
. . . . . . 7
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β π β LMod) |
24 | | imassrn 6029 |
. . . . . . . . 9
β’ (πΉ β π₯) β ran πΉ |
25 | | eqid 2737 |
. . . . . . . . . . . 12
β’
(Baseβπ) =
(Baseβπ) |
26 | 10, 25 | lmhmf 20511 |
. . . . . . . . . . 11
β’ (πΉ β (π LMHom π) β πΉ:(Baseβπ)βΆ(Baseβπ)) |
27 | 3, 26 | syl 17 |
. . . . . . . . . 10
β’ (π β πΉ:(Baseβπ)βΆ(Baseβπ)) |
28 | 27 | frnd 6681 |
. . . . . . . . 9
β’ (π β ran πΉ β (Baseβπ)) |
29 | 24, 28 | sstrid 3960 |
. . . . . . . 8
β’ (π β (πΉ β π₯) β (Baseβπ)) |
30 | 29 | adantr 482 |
. . . . . . 7
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (πΉ β π₯) β (Baseβπ)) |
31 | | inss2 4194 |
. . . . . . . . . 10
β’
(π« (Baseβπ) β© Fin) β Fin |
32 | 31 | sseli 3945 |
. . . . . . . . 9
β’ (π₯ β (π«
(Baseβπ) β© Fin)
β π₯ β
Fin) |
33 | 32 | adantl 483 |
. . . . . . . 8
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β π₯ β Fin) |
34 | 27 | ffund 6677 |
. . . . . . . . . 10
β’ (π β Fun πΉ) |
35 | 34 | adantr 482 |
. . . . . . . . 9
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β Fun πΉ) |
36 | 16 | adantl 483 |
. . . . . . . . . 10
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β π₯ β (Baseβπ)) |
37 | 27 | fdmd 6684 |
. . . . . . . . . . 11
β’ (π β dom πΉ = (Baseβπ)) |
38 | 37 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β dom πΉ = (Baseβπ)) |
39 | 36, 38 | sseqtrrd 3990 |
. . . . . . . . 9
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β π₯ β dom πΉ) |
40 | | fores 6771 |
. . . . . . . . 9
β’ ((Fun
πΉ β§ π₯ β dom πΉ) β (πΉ βΎ π₯):π₯βontoβ(πΉ β π₯)) |
41 | 35, 39, 40 | syl2anc 585 |
. . . . . . . 8
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (πΉ βΎ π₯):π₯βontoβ(πΉ β π₯)) |
42 | | fofi 9289 |
. . . . . . . 8
β’ ((π₯ β Fin β§ (πΉ βΎ π₯):π₯βontoβ(πΉ β π₯)) β (πΉ β π₯) β Fin) |
43 | 33, 41, 42 | syl2anc 585 |
. . . . . . 7
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (πΉ β π₯) β Fin) |
44 | | eqid 2737 |
. . . . . . . 8
β’ (π βΎs
((LSpanβπ)β(πΉ β π₯))) = (π βΎs ((LSpanβπ)β(πΉ β π₯))) |
45 | 17, 25, 44 | islssfgi 41428 |
. . . . . . 7
β’ ((π β LMod β§ (πΉ β π₯) β (Baseβπ) β§ (πΉ β π₯) β Fin) β (π βΎs ((LSpanβπ)β(πΉ β π₯))) β LFinGen) |
46 | 23, 30, 43, 45 | syl3anc 1372 |
. . . . . 6
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (π βΎs
((LSpanβπ)β(πΉ β π₯))) β LFinGen) |
47 | 20, 46 | eqeltrd 2838 |
. . . . 5
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β (π βΎs (πΉ β ((LSpanβπ)βπ₯))) β LFinGen) |
48 | | imaeq2 6014 |
. . . . . . 7
β’
(((LSpanβπ)βπ₯) = π΄ β (πΉ β ((LSpanβπ)βπ₯)) = (πΉ β π΄)) |
49 | 48 | oveq2d 7378 |
. . . . . 6
β’
(((LSpanβπ)βπ₯) = π΄ β (π βΎs (πΉ β ((LSpanβπ)βπ₯))) = (π βΎs (πΉ β π΄))) |
50 | 49 | eleq1d 2823 |
. . . . 5
β’
(((LSpanβπ)βπ₯) = π΄ β ((π βΎs (πΉ β ((LSpanβπ)βπ₯))) β LFinGen β (π βΎs (πΉ β π΄)) β LFinGen)) |
51 | 47, 50 | syl5ibcom 244 |
. . . 4
β’ ((π β§ π₯ β (π« (Baseβπ) β© Fin)) β
(((LSpanβπ)βπ₯) = π΄ β (π βΎs (πΉ β π΄)) β LFinGen)) |
52 | 51 | rexlimdva 3153 |
. . 3
β’ (π β (βπ₯ β (π« (Baseβπ) β© Fin)((LSpanβπ)βπ₯) = π΄ β (π βΎs (πΉ β π΄)) β LFinGen)) |
53 | 13, 52 | mpd 15 |
. 2
β’ (π β (π βΎs (πΉ β π΄)) β LFinGen) |
54 | 1, 53 | eqeltrid 2842 |
1
β’ (π β π β LFinGen) |